Abstract
In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\Bbb{R}^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a sphere or cylinder $S^{k1}\times \Bbb{R}^{nk}$. In this paper we give an affirmative answer for entire solutions in dimension $2$. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blowdown in space of any entire convex translating solution is $k$rotationally symmetric. We also prove that the blowdown in spacetime of an ancient convex solution which sweeps the whole space $\Bbb{R}^{n+1}$ is a shrinking sphere or cylinder.

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